System and method for modeling mixed signal RF circuits in a digital signal environment

ABSTRACT

A behavioral model for mixed signal RF circuits. The model approximates non-linear filtering effects for base-band (i.e. suppressed carrier) end-to-end systems analysis. The new model, the K-model, is a linear MIMO (multi-input-multi-output) model with output radius corrected by a non-linear SISO (single-input-single output) model and output angle corrected by a non-linear rotation. The SISO model uses a multi-tanh structure to synthesize a non-linear filter. The multi-tanh structure simulates non-linear behavior by gently switching between transfer functions as the base-band input varies. For excursions well into the steady state non-linear region of operation the K-model simulates large-signal base-band transients to within about 10 percent of those simulated with detailed unsuppressed-carrier models.

RELATED APPLICATION INFORMATION

This application is a continuation of U.S. application Ser. No.09/096,555, filed on Jun. 12, 1998, now U.S. Pat. No. 6,181,754 andhereby incorporated by reference as if set forth fully herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to the field of circuit simulation andmore particularly to modeling mixed signal radio frequency (RF) circuitsin a digital signal or mixed signal environment.

2. Description of Background Art

Modern communications systems are digital in nature but they requiresome analog circuitry. Unlike data traveling over traces on a chip, datatraveling between geographically different locations must ride on“carriers” that propagate over cables, air, or space. Carriers areelectromagnetic AC signals with frequencies suited for propagation overlong distances. The transmitter encodes blocks of digital data into“base-band” analog signals which then modulate a carrier. The carrierfrequency is much higher than the maximum base-band frequency.Modulation and transmission require analog circuitry. On the receiverend, analog circuitry amplifies and demodulates the carrier to recoverthe base-band signal. Analog circuitry is a crucial part of acommunications link but does not always work perfectly. Analog circuitryon either end can distort the signal. Various analog and digital signalprocessing (DSP) techniques reduce and correct distortion. Eachtechnique has benefits and costs relative to the system at hand. Theoptimal balance of DSP and analog expense is usually not obvious andhardware iterations cost a lot. The means to simulate DSP algorithms andanalog circuitry together can reduce the number of hardware iterationsby exposing-problems early.

Ideally one would simulate DSP and analog subsystems together with theanalog system modeled in detail. But should one use a DSP simulator oran analog simulator? The purely analog approach leads to impractical runtimes and requires cumbersome analog models of DSP algorithms. Since DSPsimulators cannot solve analog non-linear differential equations,simulating the analog subsystem in detail requires DSP and analogsimulators running in parallel. The parallel simulator approach iscalled “co-simulation”. The problem with co-simulation is that DSPalgorithms operate on the base-band signals while the analog circuitoperates on the carrier. The wide difference between carrier andbase-band time scales makes co-simulation extremely slow at best. Thereis another problem with co-simulation: the necessary detailed analogmodels reveal design secrets. Although vendors of analog intellectualproperty (IP) would like to supply models to potential customers they donot want to expose their IP to competitors.

The most practical alternative to co-simulation is to model the analogcircuitry for a DSP simulator. The challenge is to capture the relevantbase-band distortion in a behavioral model that is easy to extract andimplement. An extra benefit of behavioral models is that they simulatebehavior without revealing design secrets. With respect to conventionalsystems, “relevant base-band distortion” refers to linear distortion,uniform gain compression, frequency-dependent gain compression, andAM-PM conversion. The last three distortions are non-linear behaviors.

The relevant linear distortion is due to dynamics, the circuit'sdependence on input history. Models described by lineardifferential/integral equations are dynamic. In contrast, a modeldescribed by time-invariant algebraic equations is static: the presentoutput depends only on the present input; there is no dependence oninput history; the model has no memory. Typical symptoms of lineardistortion are dispersion (group delay) and intersymbol interference,things associated with a low pass filter.

Uniform gain compression refers to a loss of small signal gain when theinput has a large DC offset (a large bias). Here, “uniform” implies thegain drops by the same percentage regardless of the frequency of aninput sinusoidal perturbation. In contrast, “frequency-dependent gaincompression” refers to gain compression that varies with perturbationfrequency; gain loss still increases with increasing input bias but theloss is not uniform.

AM-PM conversion refers to a phase modulation (PM) of the output causedby an amplitude modulation of the input; the phase of the base-bandoutput varies with the amplitude of the input carrier.

What is needed is a modeling system for a mixed signal RF circuit in aDSP or mixed signal environment that (1) captures gain compression andstatic am-pm conversion, (2) captures linear distortion, (3) capturesfrequency-dependent gain compression and (4) is easy to extract andimplement.

SUMMARY OF THE INVENTION

The invention is a system and method for a behavioral model for mixedsignal RF circuits. The model approximates non-linear filtering effectsfor base-band (i.e. suppressed carrier) end-to-end systems analysis. Thenew model, the K-model, is a linear MIMO (multi-input-multi-output)model with output radius corrected by a non-linear SISO(single-input-single output) model and output angle corrected by anon-linear rotation. The non-linear SISO model uses a multi-tanhstructure to synthesize a non-linear filter. The multi-tanh structuresimulates non-linear behavior by gently switching between transferfunctions as the base-band input varies. For excursions well into thesteady state non-linear region of operation the K-model simulateslarge-signal base-band transients to within about 10 percent of thosesimulated with detailed unsuppressed-carrier models

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of a set of transfer functions for an RFreceiver.

FIG. 2 is a graphical and mathematical description of a portion of theoperation of the non-linear SISO model according to one embodiment ofthe present invention.

FIGS. 3(a)-(c) are graphical illustrations of the multi-tanhapproximation procedure according to one embodiment of the presentinvention.

FIG. 4 is an illustration of the K-model according to a preferredembodiment of the present invention.

FIG. 4a is an illustration of a technique for implementing the rotationaccording to one embodiment of the present invention

FIG. 4b is an illustration of the linear MIMO filter according to oneembodiment of the present invention.

FIG. 4c is an illustration of one embodiment of the non-linear SISOfilter.

FIG. 4d is an illustration of a piece-wise linear function that can beused in an alternate embodiment.

FIG. 5 is an illustration of the first circuit used to test thepreferred embodiment of the present invention.

FIG. 6 is an illustration of the DC transfer curve, the relationshipbetween base-band output radius and unmodulated input carrier amplituderesulting from the testing of the first circuit.

FIG. 7 is an illustration of the radial transfer functions extracted atseveral input biases resulting from the testing of the first circuitusing the present invention.

FIG. 8 is an illustration of the transfer functions normalized for unityDC gain according to the results of the testing of the first circuitusing the present invention.

FIGS. 9a and 9 b are illustrations comparing I−Q output trajectories forsmall and large in-band signals using SpectreRF model and the K-model ofthe present invention.

FIGS. 10a and 10 b are illustrations of the trajectories for small andlarge signals at higher frequencies according to the present invention.

FIG. 11 is an illustration of an RF receiver used during a second testof the present invention.

FIG. 12 is an illustration of the radial DC transfer curve using thesecond test circuit computed with SpectreRF and the K-model according tothe preferred embodiment of the present invention.

FIG. 13 is an illustration of a comparison of K-model and SpectreRFsimulations for several input levels on the second test circuitaccording to the preferred embodiment of the present invention.

FIG. 14 is an illustration comparing simulations using SpectreRF, alinear model, a cascade model, and the K-model of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A preferred embodiment of the present invention is now described withreference to the figures where like reference numbers indicate identicalor functionally similar elements. Also in the figures, the left mostdigit of each reference number corresponds to the figure in which thereference number is first used.

Wireless receivers have analog circuitry between the antenna and digitalsignal processing (DSP) components and algorithms. The analog front endcan introduce linear and non-linear distortion. Matched filters,encoders, and equalizers hopefully mitigate noise and distortion. Theoptimal balance of digital and analog complexity is not always obviousor easily specified. The only thing obvious is that “optimal” is definedin an end-to-end context. This paper describes a way to model the entireanalog front end for analysis in a DSP design environment. One objectiveis to see if an existing analog front end design will work in theend-to-end system.

There are at least four kinds of behavioral models: linear models, asdescribed in Crols and Steyaert, CMOS Wireless Transceiver Design,Kluwer Academic Publishers, 1997 which is incorporated by referenceherein in its entirety, for example, static non-linear models asdescribed in Struble et al., Understanding Linearity in WirelessCommunications Amplifiers, IEEE Journal of Solid-State Circuits,September, 1997 which is incorporated by reference herein in itsentirety, for example, cascaded models that are described in greaterdetail herein, and models based on the Volterra series as described inSchetzen, The Volterra & Wiener Theories of Non-Linear Systems, KriegerPublishing Company, 1980 which is incorporated by reference herein inits entirety, for example. Linear models simulate linear distortion overa small range of input amplitudes. A lone linear model does not exhibitgain compression. Static models scale and rotate the complex base-bandinput signal to generate base-band output. The complex plane is aconvenient space for representing base-band signals because one carrierhas two phases and each phase can be modulated independently. If thedegree of scaling and rotation depends on the magnitude of the inputbase-band signal, the static model is non-linear. Static non-linearmodels capture uniform gain compression and static am-pm conversion butnot linear distortion. One fairly obvious thing to do is to cascade astatic non-linear model with a single linear model. “Single linearmodel” means all outputs have similar impulse responses, i.e., alltransfer functions have common poles. The cascaded model produces lineardistortion, gain compression, and am-pm conversion but notfrequency-dependent gain compression.

In addition to the effects mentioned above, the K-model also simulatesfrequency-dependent gain compression. The key elements of the K-modelare a set of linear filters and a set of hyperbolic tangent functions.The K-model drives the linear filters with a multi-tanh structure asdescribed in Gilbert, The Multi-tanh Principle: A Tutorial Overview,IEEE Journal of Solid-State Circuits, January 1998 which is incorporatedby reference herein in its entirety, for example. The multi-tanhstructure was originally developed to synthesize static non-linearfunctions using transistor circuits. The multi-tanh structure inGilbert, The Multi-tanh Principle: A Tutorial Overview, IEEE Journal ofSolid-State Circuits, January 1998 was not developed as general methodfor modeling linear and non-linear distortion of complex signals.

Models based on the Volterra series capture all the effects above butrequire multi-dimensional convolution. The kernels of such convolutionsare hard to extract and the integrals are hard to implement. The K-modeluses only one-dimensional convolutions because its dynamics come from aset of linear models. One-dimensional convolutions are far easier toimplement than multi-dimensional convolutions. The general Wiener modeland Gate model, that are described in Schetzen, The Volterra & WienerTheories of Non-Linear Systems, Krieger Publishing Company, 1980 andreferenced above, are variations of the Volterra series expansion. Theyuse cascaded one-dimensional convolutions. However, the Wiener and Gatemodels are still harder to extract and implement than the K-model.

Communications systems engineers primarily use data flow simulators.When a data flow model executes, it receives inputs, generates output,then awaits the next complete set of fresh inputs. Data flow through thesystem determines when each model executes. The only possible“convergence” problem occurs when a feedback loop has no delay. The onlynotion of time comes from the sample rate. DSP algorithms typicallyoperate only on the information impressed on the RF carrier.Consequently, data flow models usually suppress RF carriers. Complexvariables represent RF signals. The real part represents the amplitudeof the “in-phase” component of the carrier and the imaginary partrepresents the amplitude of the quadrature component. For example ifrf(t) is the instantaneous RF signal and

rf(t)=a(t)cos(w _(c) t)−b(t)sin(w _(c) t)  Eq. (1)

where w_(c) is the carrier frequency, then the base-band representationis a(t)+jb(t) where

 rf(t)=Re[a(t)+jb(t))exp(jw _(c) t)]  Eq. (2)

Non-linear models of RF circuits are not new. It is relatively easy tofit a polynomial to an analytic function. Common terms like “IP3” and“IP2”, which measure intermodulation distortion, are based on polynomialmodels as described in Ha, Solid-State Microwave Amplifier Design,Krieger Publishing Company, 1991 which is incorporated by referenceherein in its entirety. Another approach is to measure the complex gainat several signal levels then interpolate on-the-fly to vary the gainwith input signals as described in Struble et al., UnderstandingLinearity in Wireless Communications Amplifiers, IEEE Journal ofSolid-State Circuits, September, 1997 which has been incorporated byreference above. Here, gain is defined as a ratio of output to input,not the ratio of incremental output to incremental input. Onedisadvantage of these first two models is that they do not filter thebase-band signal—the present output depends only on the present input.Such models have no memory and no filtering effects. Filtering effectsmake intermodulation distortion dependent on bit rate as described inHeutmaker et. al, Using Digital Modulation to Measure and Model RFAmplifier Distortion, Applied Microwave and Wireless, March/April 1997which is incorporated by reference herein in its entirety.

Memory can be introduced by cascading a memoryless non-linear model witha linear filter. However, the cascade model does not captureamplitude-dependent poles and zeroes. FIG. 1 is an illustration of a setof transfer functions for an RF receiver. Each transfer functioncorresponds to a different input signal bias. “Bias” is the amplitude ofthe unmodulated carrier. The transfer functions are defined in terms ofincremental quantities. A transfer function describes the small signalbase-band response to small signal perturbations in the amplitude of theinput RF carrier. Pole/zero movement is evident because gain compressiondepends on frequency.

Previous work on behavioral models with memory are based on the Volterraseries as described in Ha, Solid-State Microwave Amplifier Design,Krieger Publishing Company, 1991, Schetzen, The Volterra & WienerTheories of Non-Linear Systems, Krieger Publishing Company, 1980, andVerbeyst and Bossche, VIOMAP, the S-Parameter Equivalent for WeaklyNonlinear RF and Microwave Devices, IEEE Transactions on MicrowaveTheory and Techniques, Vol. 42, No.12, December, 1994 which are allincorporated by reference herein in their entirety, for example. Oneapplication of the Volterra series is to derive a frequency-dependentexpression for IP3. The Volterra series has at least three drawbackswhen applied to time-domain simulation: (a) it is difficult to extract,(b) it does not always converge, and (c) it can be as computationallyintensive as co-simulation. The Wiener model as described in Schetzen,The Volterra & Wiener Theories of Non-Linear Systems, Krieger PublishingCompany, 1980 is similar to the Volterra series except the functionalsin the expansion are orthogonal when driven by white Gaussian noise.Orthogonality circumvents some convergence problems associated with theVolterra series but does not substantially alleviate the difficultieswith extraction or implementation. In contrast, the K-model of thepresent invention is a heuristic model and is much easier to extract andimplement.

FIG. 4 is an illustration of the K-model according to a preferredembodiment of the present invention. The K-model consists of-three mainblocks: a non-linear SISO block 407, a linear MIMO block 406, and arotation block 405. Each of these blocks are described in greater detailbelow.

The K-model includes a non-linear SISO model 407. The SISO model can beunderstood by first considering a memoryless non-linearity, x(r).Instead of approximating x(r) directly we start with dx/dr. We use fuzzylogic basis functions as described in Li-Xin Wang, Adaptive FuzzySystems and Control Design and Stability Analysis, pp. 49-52, PTRPrentice Hall, 1994 which is incorporated by reference herein in itsentirety, for example, as a basis to interpolate between dx/drmeasurements, then integrate the basis functions with respect to x toobtain an approximation to x(r) The integrated basis functions are tanhcurves. The particular bases, g(r), are somewhat arbitrary. FIG. 2 is agraphical and mathematical description of a portion of the operation ofthe non-linear SISO model according to one embodiment of the presentinvention.

FIGS. 3(a)-(c) are graphical illustrations of the multi-tanhapproximation procedure according to one embodiment of the presentinvention. FIG. 3(a) is the original non-linearity shown with shortlines that represent slope measurements dx(r_(k))/dr. FIG. 3(b) showsthe weighted tanh curves. FIG. 3(c) shows the reconstructed x(r), givenby equation (3), along with the original x(r). The error is barelynoticeable.

x(r)=Σ[(c)tan h[(r−r _(k))/c]dx(r _(k))/dr]  Eq. (3)

The multi-tanh structure has been used to design programmable amplifiersas described in Gilbert, The Multi-tanh Principle: A Tutorial Overview,IEEE Journal of Solid-State Circuits, January 1998 which wasincorporated by reference supra. The present invention uses this conceptin the modeling arena. One approach is to synthesize a non-linearfilter, i.e., a non-linear element that has memory, such that thefilter's small signal transfer function matches that of the originalsystem regardless of bias. The filter is non-linear because the transferfunction changes with bias.

To synthesize a non-linear filter we replace the tangent slopes,dx(rk)/dr in equation (3), with small signal transfer functionsextracted at the appropriate bias−h_(k) is extracted with the inputbiased to r_(k).

The inverse Fourier transform of k^(th) transfer function is the impulseresponse hk(t). Multiplication of the slopes and tanh functions inequation (3) becomes convolution of the impulse responses with the tanhfunctions. Mathematically, x(r)=

ΣK=−N to N ∫h _(k)(τ)(c)tan h((r(t−τ)−r _(k))/c)dτ  Eq. (4)

where 2N+1 is the number of tanh functions, k is the summation index, τis the integration variable, r_(k) is where the k^(th) tanh functioncrosses zero, and c equals r_(N)/(2N). The tanh curves gently switchbetween transfer functions as the base-band input signal varies.

One objective of the K-model is to simulate an RF front end in a DSPdesign environment. The DSP design environment works primarily withbase-band representations of RF signals. As indicated above thebase-band representation of an RF signal is a complex number. Complexnumbers can be represented by rectangular coordinates (real andimaginary parts) or polar coordinates (radius and angle), for example.In either case the K-model of the preferred embodiment uses two inputsand two outputs. The linear MIMO filter 406 gives the K-model its basicfiltering properties in a general rectangular fashion. Rectangularcoordinates avoid the angular discontinuities caused by inputtrajectories crossing the negative real axis or diving through theorigin. However, as described infra, non-linear behavior is moreconveniently introduced in polar coordinates if one assumes the dominantnon-linearity has polar symmetry.

The equations below describe a general linear relationship between twoinputs and two outputs:

Yi(t)=K 11(t)*Xi(t)+K 12(t)*Xq(t)  Eq. (5)

Yq(t)=K 21(t)*Xi(t)+K 22(t)*Xq(t)  Eq. (6)

where “*” denotes time-domain convolution, X is the base-band input, andY is the base-band output. The linear MIMO filter 406 described in thisway is referred to as a complex filter in Crols and Steyaert, CMOSWireless Transceiver Design, Kluwer Academic Publishers, 1997,referenced above. K11, K12, K21, and K22 are elements in a matrix thatwhen post multiplied by the single column matrix [xi xq] results in thesingle column matrix [yi yq]. K11 is extracted by setting the RF carrierphase to zero, setting the signal bias to early zero, and measuring thesmall signal transfer function from modulation to “in-phase” base-bandoutput, Yi. K21 can be extracted from the same simulation by measuringthe transfer function to the “quadrature” output, Yq. K22 and K12 areextracted in the same manner but with the carrier phase set to 90degrees.

The rotation block 405 captures static AM-PM conversion as described inHa, Solid-State Microwave Amplifier Design, Krieger Publishing Company,1991, for example. In contrast, the non-linear SISO block 407 capturesAM-PM conversion due to pole zero movement. As the unmodulated inputcarrier amplitude increases, the receiver's output phase can change. Interms on the complex plane, as the input vector runs out along thepositive real axis, the output vector runs radially outward at an angle.At larger input radii, that angle can depend on input radius. If thebase-band signal is a+jb, the radius is equal to the square root of[a²+b²].

The rotation block 405 rotates the input base-band vector according tosteady state data taken at several input radii. The rotation is a staticnon-linear function of input radius. The next section describes thecomplete K-model.

In many RF receivers the dominant non-linearities depends only on inputradius. As indicated above FIG. 4, is an illustration of the K-modelaccording to a preferred embodiment of the present invention. TheK-model operates on base-band input to compute base-band output. TheK-model is a linear MIMO model with radial and rotational correctionsthat introduce non-linear behavior. Radial and rotational correctionsboth depend only on input radius.

To determine the radial correction the K-model determines the inputradius then filters it with two SISO models, one linear 408 and onenon-linear 407. Both SISO models operate on input radius and compute anoutput radius. The transfer function of the linear SISO model can be acopy of one of the transfer functions inside the non-linear SISO model.For example, it can be a copy of the transfer function extracted at thesmallest bias. The radial correction factor is the ratio of thenon-linear SISO model's output to the linear SISO model's output. Whenthe input radius is small the correction factor is nearly one. When theinput radius is large the correction factor is less than one, assumingthe non-linearity stems from saturation.

To compute the rotational correction the K-model first reads tabular Iand Q versus radius data and interpolates an I and Q depending on inputradius. The model then rotates the input by arctan (Q/I) before theinput reaches the linear MIMO filter 406. The K-model removes the DCrotation introduced by the linear MIMO filter 406 to avoid over-rotatingthe input. From an angular perspective the K-model is a cascade model, astatic non-linearity followed by a linear filter. The non-linear phasecorrection enables the K-model to simulate phase-shifts that depend oninput amplitude. The rest of this section explains each block in FIG. 4.

FIG. 4 also includes two rectangular-to-polar coordinate transformationsunits 401. One example of the operation performed by therectangular-to-polar coordinate transformations units 401 is shown inequation (7).

r=Sqrt[i ² +q ²], and q=ArcTan[q/i]  Eq. (7)

FIG. 4 also includes a polar-to-rectangular coordinate transformationsunit 402. One example of the operation performed by thepolar-to-rectangular coordinate transformation unit 402 is shown inequation (8).

i=r cos(q), and q=r sin(q)  Eq. (8)

FIG. 4 also includes a signal divider 403. The output is equal to y/x(with y and x being received as inputs). This block checks for x=0 andoutputs a set value, e.g., 1 in that event.

FIG. 4 also includes a multiplier 404, its output equals the product ofits inputs.

FIG. 4 includes the rotation block 405 which performs a rotationoperation in the complex plane. FIG. 4a is an illustration of atechnique for implementing the rotation according to one embodiment ofthe present invention. The data for the interpolation process is steadystate values of iout and qout resulting from several values of iin withqin=0. The implementation does not necessarily have to involveinterpolation. One could use a closed form expression for the phaseshift if it exists. The point is that Δθ is a non-linear function of r.One could also interchange the order of the rotation block 405 and thelinear MIMO filter 406 to correct the phase after the linear blockinstead of before it. The rotation implemented-by the correction block4055 corresponds to “θ(|ρ|)” in equation (2) of Struble et al.,Understanding Linearity in Wireless Communications Amplifiers, IEEEJournal of Solid-State Circuits, September, 1997 , that was incorporatedby reference above.

The linear MIMO filter 406 is a 2-input-2-output linear filter. FIG. 4bis an illustration of the linear MIMO filter according to one embodimentof the present invention.

FIG. 4c is an illustration of one embodiment of the non-linear SISOfilter. As described above, the non-linear SISO filter 407 determinesthe value for x(r(t)).

The preferred embodiment uses the tanh function. However, otherfunctions can also be used, e.g., other functions having an “S” shape.In an alternate embodiment a piece-wise linear function can be used,e.g., as shown in FIG. 4d.

The S-curves do not all have to be identical; one could attempt todecrease the number of curves by using curves that are different fromeach other. Also, if r0=0, symmetry can be exploited to remove thenegatively indexed filters and instead, re-use the positively indexedfilters.

The linear SISO 408 can be a copy of the h0 block illustrated-in FIG.4c.

The K-model was tested using two circuits. The first test circuit was amodel that demonstrated the need to include memory in the model. Thesecond test circuit was a larger transistor-level model thatdemonstrated the need to include frequency-dependent gain compression(pole-zero movement).

FIG. 5 is an illustration of the first circuit used to test thepreferred embodiment of the present invention. The test circuittranslates base-band inputs up to RF, e.g., 1 Ghz, passes them through asaturating pass-band active filter, then downconverts to base-band. Theamplifier saturates at 10 volts when the input exceeds 1 volt. Theband-width of the circuit is 70 Mhz and the carrier is 1 Ghz.

Recall, that “bias” refers to the amplitude of the unmodulated 1 Ghzcarrier. The small-signal input, for which transfer functions aredefined, is an infinitesimal modulation of the carrier amplitude. FIG. 6is an illustration of the DC transfer curve, the relationship betweenbase-band output radius and unmodulated input carrier amplituderesulting from the testing of the first circuit using the preferredembodiment of the present invention.

FIG. 7 is an illustration of the radial transfer functions (i.e.input-to output radius) extracted at several input biases resulting fromthe testing of the first circuit using the present invention. FIG. 7shows gain compression. FIG. 8 is an illustration of the transferfunctions normalized for unity DC gain according to the results of thetesting of the first circuit using the present invention. FIG. 8 showspole-zero movement since the normalized transfer functions do not allcoincide.

We compared K-model simulations of the first test circuit to simulationsperformed using SpectreRF that is commercially available from CadenceDesign Systems, Inc., San Jose, Calif. We used the a DSP tool, SPW(Signal Processing Workstation) which is commercially available fromCadence Design Systems, Inc., to run the K-model. We compared I−Q (i.e.in-phase versus quadrature) trajectories since the systems engineer mayvisualize distortion in terms of a symbol constellation, which lies inthe I−Q plane. FIGS. 9a and 9 b are illustrations comparing I−Q outputtrajectories for small and large in-band signals using SpectreRF modeland the K-model of the present invention. The small signal SpectreRF andK-model responses coincide. The large signal K-model response is about 8percent smaller than the SpectreRF response; if we were to scale theK-model's response by 1.08 the outer edges of the trajectories wouldcoincide. The DC transfer curve starts to saturate at an input voltageof 1 volt. The combined base-band input signals produced a peak inputsignal of 1.82 volts.

FIGS. 10a and 10 b are illustrations of the trajectories for small andlarge signals at higher frequencies, (inphase=70 Mhz and quadrature=150Mhz) according to the present invention. Both input signals were 1 voltsinusoids, giving a combined peak base-band signal of 1.4 volts. Theresponse is significantly better than a memoryless model. The aspectratio of the overall response is much different than one because theresponse depends on frequency and the in-phase and quadrature inputs areof different frequencies. A memoryless model could not produce thecorrect aspect ratio because it would apply the same attenuation to bothaxes. A memoryless model would produce an aspect ratio of one

A second test was also performed that compared K-model and SpectreRFsimulations for a large transistor level model of an RF receiver. FIG.11 is an illustration of an RF receiver used during a second test of thepresent invention.

FIG. 1 illustrates the radial transfer function at several input signalbiases. Note the difference in gain compression at DC and 10 Mhz. Thedifference is due to amplitude-dependent poles and zeros. FIG. 12 is anillustration of the radial DC transfer curve using the second testcircuit computed with SpectreRF and the K-model according to the presentinvention. The output is well into saturation when the unmodulatedcarrier exceeds 20 mv.

FIG. 13 is an illustration of a comparison of K-model and SpectreRFsimulations for several input levels on the second test circuitaccording to the present invention. The input in-phase and quadraturesignals were 5 Mhz and 10 Mhz respectively and of equal amplitude. Thecombined input signals gave peaks of 1.25 mv, 0.025 mv, and 30 mv. Basedon the DC transfer curve the latter two input test signals were large.The error in the large signal response was on the order of 10 percent.

Trajectories from a cascade model did not significantly differ fromthose in FIG. 13. One might be tempted to use the cascade model.However, while it is important to simulate large signal swings it isalso be important to simulate small perturbations about a largedisplacement. An I−Q trajectory may spiral into a constellation pointinstead of jumping right to it. The next test addresses the latterscenario.

FIG. 14 is an illustration comparing simulations using SpectreRF, alinear model, a cascade model, and the K-model of the present invention.For each simulation the input was a small constant amplitude sinusoid,10 Mhz away from the carrier, added to a large unmodulated carrier. Inthe input complex plane the input signal traced out a small circlecentered on the real axis and displaced from the origin. SpectreRFproduced the lower oval. The uncorrected linear MIMO model within theK-model produced the largest trajectory in FIG. 14. Alone, the linearmodel is way off because it does not account for saturation. Thecomplete K-model produced the upper oval. A cascade model produced thethin vertical trajectory. The cascade model in this example consisted ofa memoryless non-linear multi-tanh structure followed by a normalizedlinear MIMO model. The memoryless multi-tanh structure used the sametanh curves used in the K-model but the associated slopes were just theDC values of the K-model's transfer functions. The memoryless multi-tanhstructure operated only on radius. The resulting radius was combinedwith the input phase to convert back to rectangular coordinates anddrive the linear MIMO model 406. The linear MIMO model 406 was theK-model's linear MIMO model normalized for a radial gain of one.

In the radial direction the cascade model does not account for the polesand zeros that depend on radius. In general, cascade models cannotsimulate pole-zero movement because all filtering effects come from asingle linear MIMO model. The non-linear part of the cascade model canscale the linear filter's response depending on bias but it can not movethe poles and zeros. In contrast, the K-model can move the poles andzeros for bias.

The K-model is an improvement over linear models, memoryless non-linearmodels, and cascaded combinations of linear filters and memorylessnon-linearities. Simulations of two different test circuits show themodeling error is about 10 percent for input excursions well into thesaturation region of the DC transfer curve.

The first test circuit composed mostly of behavioral blocks implementedin analog hardware description language (AHDL) code. The second testcircuit was composed of device models and used very little AHDL. Asidefrom validating the K-model, the two test circuits demonstrate that theK-model can be used to extract a system-level model from any mix ofbehavioral and device models Accordingly the K-model can be used duringany phase of the receiver's design to see if the current design willwork in the larger system.

As indicated above SPW was used to simulate the K-model. SPW read intransfer functions extracted using SpectreRF then applied an inversefast fourier transform (IFFT) to compute impulse responses. To simulatea transfer function, SPW convolved the sampled input with the sampledimpulse response. There are other ways to simulate the transferfunctions. One could apply the FFT to chunks of input data, multiply theFFT by the transfer function, then apply an IFFT to return to the timedomain. However, that approach does not work well in feedback loopsbecause the output is delayed by the length of a data chunk. Anotherapproach is to fit rational transfer functions to the measured transferfunctions. The rational transfer functions can be decomposed intopole-residue forms which can then be simulated with recursiveconvolution one example of which is described in Lin and Kuh, TransientSimulation of Lossy Interconnects Based on the Recursive ConvolutionFormulation, 39 IEEE Trans. Circuits Syst. 11, pp 879-892 (1992) that isincorporated by reference herein in its entirety. Recursive convolutionis very fast. The first step is to make sure the rational transferfunction approach is as good as or better than the finite impulseresponse (FIR) filter approach.

Along with increased simulation speed the rational transfer functionapproach may enable the K-model to simulate very low frequency dynamics.The automatic gain control (AGC) loop for example introduces very lowfrequency dynamics. An FIR filter can not simulate transients lastinglonger than the number of taps times the sample period. The rationaltransfer functions would be implemented as IIR filters, which can havetransients of arbitrary duration.

While the invention has been particularly shown and described withreference to a preferred embodiment and several alternate embodiments,it will be understood by persons skilled in the relevant art thatvarious changes in form and details can be made therein withoutdeparting from the spirit and scope of the invention.

What is claimed is:
 1. A system for modeling an output signal producedby mixed analog and digital circuitry in a digital signal designenvironment that provides an input signal, the input and outputcoordinates and the input and output signals having a radial componentand an angular component when represented in polar coordinates, thesystem comprising: a non-linear static angular correction unit, disposedto receive the input signal, for rotating the input signal to generate arotated signal; a first linear filter, disposed to receive the rotatedsignal, to linearly filter said rotated signal to generate a filteredrotated signal; a non-linear dynamic radial correction unit comprising:a second linear filter, disposed to receive said radial component of theinput signal, to linearly filter said radial component of the inputsignal and to generate a first radial correction component; a non-linearfilter, disposed to receive said radial component of the input signal,to generate a second radial correction component; a radial correctionfactor unit, comprising a signal divider and disposed to receive saidfirst and second radial correction components, to determine a radialcorrection factor based upon said first and second radial correctioncomponents; and a correction unit, for integrating said radialcorrection factor and said filtered rotated signal to model said outputsignal.
 2. The system of claim 1, wherein said correction unitintegrates said radial correction factor and said filtered rotatedsignal by multiplying together said radial correction factor and saidfiltered rotated signal.
 3. The system of claim 1, wherein said anon-linear static angular correction unit uses said radial component ofthe input signal to determine a rotation for said rotated signal.
 4. Thesystem of claim 1, wherein said non-linear filter generates said secondradial correction component using approximately S-shaped basisfunctions.
 5. The system of claim 4, wherein said approximately S-shapedbasis functions comprise hyperbolic tangent functions.
 6. A method ofmodeling an output signal produced by mixed analog and digital circuitryas part of a digital signal design that provides an input signal, theinput and output signals having an in-phase component and a quadraturecomponent when represented in rectangular coordinates and the input andoutput signals having a radial component and an angular component whenrepresented in polar coordinates, the method comprising the steps of:(a) rotating and linearly filtering the input signal to generate afiltered rotated signal; (b) linearly filtering said radial component ofthe input signal to generate a first radial correction component; (c)non-linearly filtering said radial component of the input signal togenerate a second radial correction component; (d) determining a radialcorrection factor based upon said first and second radial correctioncomponents; and (e) integrating said radial correction factor and saidfiltered rotated signal to model said output signal.
 7. The method ofclaim 6, wherein step (d) is performed by dividing said second radialcorrection component by said first radial correction component.
 8. Themethod of claim 6, wherein step (e) is performed by multiplying togethersaid radial correction factor and said filtered rotated signal.
 9. Themethod of claim 6, wherein the input signal is rotated using said radialcomponent of the input signal to determine a rotation for said rotatedsignal.
 10. The method of claim 6, wherein step (c) is performed usingapproximately S-shaped basis functions.
 11. The method of claim 10,wherein said approximately S-shaped basis functions comprise hyperbolictangent functions.
 12. The method of claim 6 further comprising thesteps of: (f) converting the model of said output signal into a transferfunction represented by a set of data points; (g) instantiating thetransfer function into the digital signal design; and (h) simulating thedigital signal design.
 13. A computer-assisted method of modeling anoutput signal produced by mixed analog and digital circuitry as part ofa digital signal design that provides an input signal, the input andoutput signals having an in-phase component and a quadrature componentwhen represented in rectangular coordinates and the input and outputsignals having a radial component and an angular component whenrepresented in polar coordinates, the method comprising the steps of:(a) rotating and linearly filtering the input signal to generate afiltered rotated signal; (b) linearly filtering said radial component ofthe input signal to generate a first radial correction component; (c)non-linearly filtering said radial component of the input signal togenerate a second radial correction component; (d) determining a radialcorrection factor based upon said first and second radial correctioncomponents; and (e) integrating said radial correction factor and saidfiltered rotated signal into a model of said output signal.
 14. Themethod of claim 13, wherein step (d) is performed by dividing saidsecond radial correction component by said first radial correctioncomponent.
 15. The method of claim 13, wherein step (e) is performed bymultiplying together said radial correction factor and said filteredrotated signal.
 16. The method of claim 13, wherein the input signal isrotated using said radial component of the input signal to determine arotation for said rotated signal.
 17. The method of claim 13, whereinstep (c) is performed using approximately S-shaped basis functions. 18.The method of claim 17, wherein said approximately S-shaped basisfunctions comprise hyperbolic tangent functions.
 19. The method of claim13 further comprising the steps of: (f) converting the model of saidoutput signal into a transfer function represented by a set of datapoints; (g) instantiating the transfer function into the digital signaldesign; and (h) simulating the digital signal design.